"Let's try kimchi."
Basically adding 아/어/여보다 to a verb in the way it's done here gives it the meaning "to try to ____ (the original verb)" For example: 이것은 아직 안 먹어봤어요 - "I haven't tried to eat this" as in "I haven't tried this food".
A good way to remember this is that 보다 is "to see", so you can think of it as trying something "to see if you like it"! I don't think that has anything to do with the actual reason 보다 is used, but it might be helpful for remembering.
보자 is the propositive/hortative conjugation of 보다 at 해라체 ("written speech level") as opposed to the formal speech level of the given sentence.
This speech level is typically reserved for written Korean: it is a formal, non-polite speech level. When spoken, it is considered forceful when talking to your social equals and inferiors and rude when talking to your social superiors.
I believe the correct verb conjugation to use is "드셔보십니다"
먹다 → 드시다 → 드셔보다 → 드셔보시다 → 드셔보십니다